• # question_answer DIRECTION (Qs. 82): Each of these questions contains two statements: Statement-1 (Assertion) and Statement-2 (Reason). Choose the correct answer (ONLY ONE option is correct) from the following- Let $x,\,\,\,y,\,\,\,z$ are three integers lying between $1$ and $9$ such that $x\,51,\,\,\,y\,41$ and $z\,31$ are three digit numbers. Statement-1: The value of the determinant$\left| \begin{matrix} 5 & 4 & 3 \\ x\,51 & y\,41 & z\,31 \\ x & y & z \\ \end{matrix} \right|is\,\,zero$. Statement-2: The value of a determinant is zero if the entries in any two rows (or columns) of the determinant are correspondingly proportional. A)  Statement-1 is false, Statement-2 is true.B)  Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.C)  Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.D)  Statement-1 is true, Statement-2 is false.

$\Delta =\left| \begin{matrix} 5 & 4 & 3 \\ x\,51 & y\,41 & z\,31 \\ x & y & z \\ \end{matrix} \right|$         $=\left| \begin{matrix} 5 & 4 & 3 \\ 100x+51 & 100y+41 & 100z+31 \\ x & y & z \\ \end{matrix} \right|$ $=\left| \begin{matrix} 5 & 4 & 3 \\ 1 & 1 & 1 \\ x & y & z \\ \end{matrix} \right|$   $[{{R}_{2}}\to {{R}_{2}}-100{{R}_{3}}-10{{R}_{1}}]$ which is zero provided$x,\,\,y,\,\,z$are in$A.P.$